Stability of Tails and 4-Canonical Models

We show that the GIT quotients of suitable loci in the Hilbert and Chow schemes of 4-canonically embedded curves of genus $g\ge 3$ are the moduli space $\bar{M}_g^{\text{ps}}$ of pseudo-stable curves constructed by Schubert in \cite{Schubert} using Chow varieties and 3-canonical models. The only new ingredient needed in the Hilbert scheme variant is a more careful analysis of the stability with respect to a certain 1-ps $\lambda$ of the $m^{\text{th}}$ Hilbert points of curves $X$ with elliptic tails. We compute the exact weight with which $\lambda$ acts, and not just the leading term in $m$ of this weight. A similar analysis of stability of curves with rational cuspidal tails allows us to determine the stable and semistable 4-canonical Chow loci. Although here the geometry of the quotient is more complicated because there are strictly semi-stable orbits, we are able to again identify it as $\bar{M}_g^{\text{ps}}$. Our computations yield, as byproducts, examples of both $m$-Hilbert unstable and $m$-Hilbert stable $X$ that are Chow strictly semi-stable.